North Carolina Democratic Gov. Roy Cooper vetoed a controversial bill that would ban most abortions after 12 weeks, setting up a likely override effort from the state legislature, where Republicans have a supermajority.

Cooper's swift veto comes just over a week after the Republican-controlled state Senate advanced the bill to his desk in a party-line vote.

Speaking to a crowd of supporters at a rally in Raleigh before his veto, Cooper urged those gathered to put pressure on four state Republican lawmakers who had previously vowed to protect abortion rights to stand by their comments and not join any veto override.

"We are going to have to kick it into an even higher gear when that veto stamp comes down. If just one Republican in either the House or the Senate keeps a campaign promise to protect women's reproductive health we can stop this ban," Cooper said at the Saturday rally. "But that's going to take every single one of you to make calls, to send emails, to write letters. Tell them to sustain this veto. Tell them to ask the Republican leadership to stop it."

Ahead of the veto, Cooper had moved to escalate pressure on the four Republicans -- state Reps. Ted Davis, John Bradford, Tricia Cotham and Republican State Sen. Michael Lee -- by paying visits to their districts. All four previously vowed to protect abortion rights and held roundtable discussions there on the topic in the hopes of appealing to those lawmakers and their constituents.

CNN has reached out to the lawmakers for comment. Davis declined to comment on his plans, while the others have not responded.

If both chambers approve the override by three-fifths majority, the bill becomes law. If it does not get the votes, then the governor's veto remains.

Demi Dowdy, a spokesperson for Republican House Speaker Tim Moore, told CNN she is confident lawmakers will override Cooper's veto and dismissed his efforts to "press these Republican members into voting his way."

Republicans in the state gained a veto-proof supermajority last month when then-Democratic Cotham switched parties. Cotham, who had campaigned in favor of abortion rights as a Democrat, was one of the Republicans who voted for the bill. She also sponsored legislation earlier this session to codify Roe v. Wade.

State legislators have already shown they are not afraid to use their newfound powers to override Cooper's vetoes. In March, the state legislature overrode Cooper's veto on a bill that now makes it legal to purchase a pistol without a permit in the state.

North Carolina had been a haven for women seeking abortion care as Southern states tightened restrictions after the Supreme Court sent the issue of abortion rights back to the states.

If Cooper's veto is not sustained, the bill -- Senate Bill 20, dubbed the "Care for Women, Children and Families Act" -- would ban any licensed physician from performing surgical abortions after the 12th week of pregnancy. It would provide exceptions in the case of rape and incest through 20 weeks of pregnancy or in the event of a "life-limiting anomaly" through 24 weeks.

Under the bill, the procedure could be performed by a physician if a doctor determines an abortion is necessary to avert death -- "not including psychological or emotional conditions."

Under current state law, abortions are legal up to 20 weeks of pregnancy. A reduction to 12 weeks would be significant but less restrictive than some other GOP-led states that have recently enacted six-week or near-total bans on the procedure.

The bill would also prohibit health care providers who objects to abortion "on moral, ethical, or religious grounds" from being required to participate in medical procedures that would result in an abortion.

It also calls for in-person examination by physicians seeking to prescribe or administer abortion-inducing drugs and would prohibit people within the state from mailing such drugs to a pregnant woman. The bill would mandate that doctors confirm that the "probable gestational age" of a fetus is no more than 10 weeks before they can sign off on a medication abortion.

This story has been updated with additional information.